Newsvendor Bounds and Heuristics for Serial Supply Chains with Regular and Expedited Shipping

Transcription

1 Newsvendor Bounds and Heursts for Seral Supply Chans wth egular and xpedted Shppng Sean X. Zhou, Xul Chao 2 Department of Systems ngneerng and ngneerng Management, The Chnese Unversty of Hong Kong, Shatn, New Terrtores, Hong Kong 2 Department of Industral and Operatons ngneerng, Unversty of Mhgan, Ann Arbor, Mhgan eeved 3 Otober 2007; revsed 20 September 2009; aepted 2 Otober 2009 DOI 0.002/nav Publshed onlne 9 November 2009 n Wley InterSene Abstrat: We study an nfnte-horzon, N-stage, seral produton/nventory system wth two transportaton modes between stages: regular shppng and expedted shppng. The optmal nventory poly for ths system s a top down ehelon base-stok poly, whh an be omputed through mnmzng 2N nested onvex funtons reursvely Lawson and Porteus, Oper es , In ths artle, we frst present some strutural propertes and omparatve stats for the parameters of the optmal nventory poles, we then derve smple, newsvendor-type lower and upper bounds for the optmal ontrol parameters. These results are used to develop near optmal heurst solutons for the ehelon base-stok poles. Numeral studes show that the heurst performs well Wley Perodals, In. Naval esearh Logsts 57: 7 87, 200 Keywords: mult-ehelon systems; ehelon base-stok poly; bounds and heursts; regular and expedted shppng. INTODUCTION Dynam leadtme management n supply hans an hedge aganst market flutuatons and effetvely balane nventory and ustomer demand. To aheve flexblty n leadtmes n a supply han, a wdely adopted strategy s the use of multple transportaton modes. Although a replenshment wth shorter leadtme an better respond to ustomer demand, t s usually more ostly. It s, therefore, mportant for the ompany to strategally determne the shppng quanttes usng dfferent leadtme and ost ombnatons based on the nventory status to mnmze ts total operatonal osts. In ths artle, we onsder a perod-revew seral supply han wth N stages. ah stage replenshes ts nventory from ts mmedate upstream stage and random ustomer demand ours at the most downstream stage. xess demand n eah perod s baklogged. Two modes of transportaton are avalable between any adaent stages, regular shppng and expedted shppng, and transportaton leadtmes are and 0, respetvely an extenson to a more general leadtme settng s dsussed n Seton 5. The expedted shppng ost s hgher than the regular shppng ost. The zero leadtme allows the frm to shp produts from any upstream stage to Correspondene to: S.X. Zhou zhoux@se.uhk.edu.hk most downstream stage n no tme, f needed, by usng expedted shppng between stages. There s a lnear holdng ost at eah stage, and a lnear shortage ost at stage when a baklog of ustomer demand ours. The obetve s to mnmze the total dsounted ost of the system over an nfnte plannng horzon. Ths problem has been studed by Lawson and Porteus [3], who demonstrated that a top down ehelon base-stok poly was optmal see also ef. 4. A top down ehelon basestok poly s haraterzed by two base-stok levels for eah stage, one for expedted orderng and the other for regular orderng. These optmal ehelon base-stok levels an be obtaned by solvng 2N nested sngle-dmensonal onvex optmzaton problems reursvely. Although the algorthm tself s qute smple, omputaton remans a tedous proess, and ts omplexty nreases wth the number of stages. Ths motvates us to develop smple newsvendor bounds and heursts for the optmal poles of eah stage of the multehelon system, whh not only an nrease ther mplementablty but also shed lght on the effet of system ontrol parameters. Ths artle makes two man ontrbutons to the lterature. Frst, we provde several mportant strutural results for the optmal expedted and regular base-stok levels. These fndngs ould advane our understandng of the system and the 2009 Wley Perodals, In.

2 72 Naval esearh Logsts, Vol propertes of the optmal poles. Seond, based on these strutural propertes, we develop three sets of newsvendor upper bounds and three sets of newsvendor lower bounds for the optmal ehelon base-stok levels for both the regular and expedted shppng modes. These bounds an be easly alulated from the system parameters. A smple heurst s then onstruted based on these bounds to ompute the nearoptmal base-stok levels. Numeral studes show that the heurst performs well. The rest of the artle s organzed as follows. In Seton 2, we revew the fndngs of prevous studes n ths feld. In Seton 3, we explan the model formulaton and present some prelmnary results. We provde a omputatonal algorthm for the optmal base-stok levels and gve some strutural results. In Seton 4, we derve several sets of lower bounds and upper bounds for the optmal ehelon base-stok levels. In Seton 5, we extend the results to a ase wth more general leadtmes. In Seton 6, we develop a smple heurst and test ts effetveness by numeral studes. We onlude the artle wth a few remarks n Seton 7. The omtted proofs are provded n the Appendx. Throughout the artle, we use the terms expedted order and expedted shppng nterhangeably. We also use the terms nreasng and dereasng n a weak sense, as representng non-dereasng and non-nreasng, respetvely. A s the ndator funton takng value f A s true, and 0 otherwse. For any real numbers a and b, a b = mn{a, b}, a b = max{a, b}. For any realvalue, monotone funton f, we use f to represent ts nverse funton. Whenever possble, we follow the notaton of Lawson and Porteus [3]. 2. LITATU VIW The body of researh related to ths work an be dvded nto two maor ategores. The frst analyzes ontrol poles for sngle-stage and mult-stage nventory models wth multple transportaton modes, and the seond develops smple bounds and heursts for the optmal ontrol parameters. We revew the varous studes n these two ategores separately. The earlest study of nventory models wth two delvery modes was made by Barankn [], who studed a sngle-perod problem. Danel [5] was the frst to onsder a mult-perod sngle-stage model wth two shppng modes. The leadtmes of regular shppng and emergeny shppng were and 0, respetvely. Fukuda [0] extended Danel [5] to the ase where the leadtmes of the two supply modes were L and L +, respetvely, for a general non-negatve value of L. Whttemore and Saunders [20] onsdered the dual-suppler problem wth leadtmes of arbtrary length, and demonstrated that the optmal ontrol poly was very omplated and state-dependent f the dfferene n leadtmes was greater than. Beause of the omplexty of systems wth general leadtmes, Sheller-Wolf et al. [5] and Veeraraghavan and Sheller-Wolf [9] foused on the evaluaton and optmzaton of two lasses of heurst poles, vz., sngle ndex and dual ndex poles. eently, Sheopur et al. [7] have shown that the lassal lost sales nventory problem s a speal ase of the dual supply modes problem. They also proposed two lasses of heurst poles and showed that one of them provded an average ost savng of.% over the best dual ndex poly of Veeraraghavan and Sheller-Wolf [9]. All these studes have foused solely on sngle-stage nventory systems. Other related work on sngle-stage nventory systems wth multple transportaton modes has been done by Feng et al. [8, 9] and Song and Zpkn [8]. For mult-ehelon models wth the opton of expedted shppng between stages, Lawson and Porteus [3] onsdered both fnte-horzon and nfnte-horzon seral systems wth dual transportaton modes. Under the assumptons that the leadtmes for regular and expedted shppng between any two adaent stages were and 0, respetvely, and that shppng osts were addtve lnear, they establshed the optmalty of a top down ehelon base-stok poly. For suh a poly, the ontrol parameters of eah ehelon onsst of two numbers, one for regular shppng and the other for expedted shppng. Muharremoglu and Tstskls [4] extended the model of Lawson and Porteus [3] to a more general settng by usng the unt analyss approah. Under a supermodular ost struture on expedted and regular shppng, they haraterzed the optmal poly as an extended ehelon base-stok type. When there s only one transportaton mode, the system of Lawson and Porteus [3] s redued to the lassal Clark Sarf model, whh has been extensvely studed, notably by Clark and Sarf [4], Federgruen and Zpkn [7], and Chen and Zheng [3]. Several studes on smple bounds of ost and optmal poles for the Clark Sarf model have been reported. allego and Zpkn [] dsussed the ssue of stok postonng and onstruted three heursts to alulate the average ost for seral produton-transportaton systems. Zpkn [2] ntrodued a lower bound for a two-stage system by restrtng the possblty of holdng nventory at the upper stream stage. Dong and Lee [6] developed a lower bound for optmal poles of nfnte-horzon seral systems wth dsounted ost rteron. For average ost rteron, Shang and Song [6] obtaned smple newsvendor-type of bounds and developed smple heursts usng a dfferent approah from that of Dong and Lee [6]. More reently, Chao and Zhou [2] have adopted another approah, onstrutng bounds and heursts for seral systems that work for both dsounted and average osts, and obtanng a seres of bounds for optmal base-stok levels. A related study on bounds and heursts for seral systems has been made by allego and Özer [2]. Naval esearh Logsts DOI 0.002/nav

3 Zhou and Chao: Newsvendor Bounds and Heursts 73 Comparng the analyss of ths artle to that n Chao and Zhou [2] for the Clark Sarf model wth an nfnte horzon, we note the followng two man dfferenes. Frst, due to the exstene of expedted orderng, eah stage s regular order deson n eah perod depends on ts expedted order deson of the next perod, whh n turn ntertwnes wth ts downstream stage s regular order deson. However, n the Clark Sarf model, the deson n one perod does not depend on that of the followng perod sne the ehelon base-stok level s myop and statonary. Seond, our system nurs two types of ndued penalty ost. One ours between adaent stages the ost harged from regular order of stage to stage + due to the nsuffent stok at stage +, and the other wthn eah stage the ost harged from expedted order to regular order of stage due to nsuffent stok from the regular order, whereas the Clark Sarf model features only one ndued penalty ost between stages. Our analyss also shows that the expedted opton between stages annot be smply onsdered as an addtonal stage n the Clark Sarf model. These two dfferenes make the dervaton of dstrbuton-funton solutons and smple lower and upper bounds muh more omplated and hallengng. 3. TH MODL AND STUCTUAL SULTS Consder an nfnte-horzon, perod-revew seral nventory system wth N stages, ndexed by, 2,..., N. Customer demand ours at stage. Stage obtans supples from stage 2, stage 2 from stage 3, and so on, and stage N replenshes ts nventory from an outsde soure stage N + wth an ample supply. Unsatsfed demand s fully baklogged at stage. Demands n dfferent perods are ndependent and dentally dstrbuted..d. non-negatve random varables. ah stage has two orderng desons: expedted order and regular order. For eah stage, the leadtme l r for regular order s, and l e for expedted order s 0 an extenson that l e = l and l r = l + for a general non-negatve l s dsussed n Seton 5. The assumpton that the leadtme dfferene between a regular and an expedted order must be appears restrtve, but relaxng t makes the problem too omplated to yeld an optmal ontrol poly that s analytally solvable. Ths s beause the resultng state spae for eah stage has to be augmented to nlude the ppelne nventory sheduled to arrve n future perods. In ths ase, t s known that, even for a sngle-stage system N =, the optmal poly s omplated and state-dependent Whttemore and Saunders [20]. For eah stage, the unt expedted and regular order ost from stage + sk and k, respetvely, wth k >k. helon holdng ost h s nurred for eah unt of on-hand nventory held n ehelon per perod, whereas baklog ost p s nurred for eah unt of baklog at stage per perod. The nstallaton holdng ost for stage s H = N = h. The followng addtonal notaton s needed. D = -perod demand -fold onvoluton of sngleperod demand as demands are..d. aross perods, =, 2,...; F = the umulatve dstrbuton funton of D, =, 2,...; F = F, =, 2,...; α = the dsount fator,.e., 0 α<. For notatonal smplty, we use D to denote a gener one-perod demand and suppress the subsrpt of F and F when =,.e., F = F and F = F. The sequene of events s as follows. Frst, at the begnnng of every perod, eah stage reeves the regular order plaed n the prevous perod. Seond, startng from stage N, eah stage plaes expedted and regular orders sequentally. Spefally, stage N frst plaes ts expedted order from the outsde suppler and reeves t mmedately. It then plaes a regular order whh wll be delvered at the begnnng of next perod note that stage N s expedted order s mmedately avalable to satsfy the order from stage N. Stage N then dedes ts expedted and regular orders from stage N. Agan, the expedted order stage N reeved an be used to satsfy the order from stage N 2. Ths top down orderng proess ontnues untl stage plaes ts expedted and regular orders from stage 2. As an expedted order has a shppng leadtme 0, the model allows expedtng from any upstream stage to any downstream stage subet to nventory avalablty at the upstream stages. Fnally, demand s realzed durng the perod at stage and all osts are nurred at the end of the perod. The obetve s to mnmze the total dsounted ost over an nfnte plannng horzon. Lawson and Porteus [3] showed that the optmal poly of ths problem s of a base-stok type, and that the optmal base-stok levels an be omputed through a nested reursve algorthm. Frst, let = k k + h and = αk k. Beause k >k and h 0, > 0. In addton, we assume 0. If ths s not the ase, then the regular shppng mode wll never be used and the model redues to the Clark Sarf model wth a sngle supply mode between stages. To see that, suppose < 0orαk <k, then t s more eonomal for stage to order the unt usng expedted shppng n the next perod rather than to order t at the urrent perod usng a regular order. Lawson and Porteus [3] defned = k αk 0, but here we keep the ost oeffent non-negatve. We all the relatve unt expedted orderng ost and the relatve unt regular orderng ost n the rest of the artle, relatve s oasonally skpped for the sake of smplty. Lawson and Porteus [3] also onsdered a so-alled unt detaned ost at stage, whh s assumed to be 0 n ths artle for the ease of exposton. Our results an easly be extended to nlude suh a ost, f desred. Naval esearh Logsts DOI 0.002/nav

4 74 Naval esearh Logsts, Vol We now present the omputatonal algorthm for the optmal base-stok levels. Let x = max{ x,0}. Defne y = y + H + p[y D ], whh s onvex wth mnmzer s. Let, for =,..., N, y = y + y s + α [ ] y D s, s = arg mn y y, 2 and, for =,..., N, + y = + y + y s, 3 s+ = arg mn y + y, 4 where both and are unvarate onvex funtons. Here, we refer to y s and y s as the ndued penalty funton wthn stage and ndued penalty funton between stages and +, respetvely. 2 Note that the algorthm above does not dretly lead to the mnmum total dsounted ost for the system. The optmal top down ehelon base-stok poly works as follows see ef. [3]. Startng from stage N, eah stage tres to rase ts ehelon nventory level and poston to the expedted base-stok level s and regular base-stok level s, respetvely, takng upstream desons as gven and gnorng downstream desons. Comparng 4 wth the optmzaton algorthm of the Clark Sarf system see some detals n Seton 5, we note the followng mportant dfferenes. Frst, for eah stage, onsder the regular base-stok level s as the base-stok level n the Clark Sarf model. To ompute s, we need to frst optmze an addtonal expedted orderng base-stok level s usng 4. Seond, due to the exstene of two base-stok levels for eah stage, the endng nventory level of a perod ould be hgher than ts expedted base-stok level n the next perod. Ths results n the thrd term n, whh s not n the optmzaton algorthm of the Clark Sarf model. It shows that the regular order deson of the urrent perod depends on the expedted order deson of the next perod, whh then n turn nfluenes ts downstream stage s regular order. Thus, n the strt sense, the optmal poly s not myop but one-perod ahead and the expedted order deson annot be smply regarded as an addtonal stage between two ehelons of the Clark Sarf model. In the rest of ths seton, we present some strutural propertes on the optmal poles that wll be used to derve smple bounds for the optmal base-stok levels. 2 A more prese defnton of the ndued penalty funtons should be y s s and y s s ; but sne s and s are onstant and wll not affet the analyss, they are omtted. POPOSITION : s <s, for = 2,..., N. s <s, for =,..., N. POOF: For part, note that s s determned by y = + y s = 0. 5 When y = s, + s = > 0. Thus, t follows from the onvexty of y that s <s. Smlarly, for part, note that s s the soluton of y = + y s + α [ ] y D s = 0. 6 Sne, for y s, y s 0 and [ y D s ] = 0, we have y < 0as > 0. The onvexty of y mples s >s. In the followng paragraphs, we develop solutons for the optmal base-stok levels s and s, whh solely depend on the demand dstrbuton, whh we term dstrbuton-funton solutons. By qs. 4, the optmal base-stok levels s and s are the soluton of y = 0 and y = 0, respetvely. For stage, takng dervatve of y wth respet to y yelds y = H + ppd > y = 0, 7 hene the optmal expedted base-stok level for stage s s = F. 8 H + p Note that f >H +p, then y > 0 from 7, whh mples that s = and the expedted shppng s never used at stage. So, n general for =, 2,..., we defne F x = for x> F x = for x<0 and smlarly F x = for x<0 F x = for x>. To solve s, t follows from part of Proposton that we only need to onsder the soluton of y = 0on y s. Thus, s s the soluton of y = + α P D y s αh + pp D y s, D2 >y = 0, 9 n whh D2 = D + D and D s another sngle-perod demand ndependent of D. For a sngle-stage system wth two delvery modes, qs. 8 and 9 provde the optmal solutons, whh an be easly omputed. Naval esearh Logsts DOI 0.002/nav

5 Zhou and Chao: Newsvendor Bounds and Heursts 75 Applyng 9 and Proposton, we obtan that s2 s the soluton of 2 y = 2 + H + ppd > y y<s + α P D y s αh + p P D y s, D2 >y = 0, 0 and that s2 s the soluton of 2 y = 2 + α 2 P D y s2 α D y s 2, D>y s + α P D y s2, D>y s + α 2 P D y s2, D>y s, D2 y s αh + pp D y s2, D>y s, D2 >y α 2 H + pp D y s2, D>y s, D2 y s, D3 >y = 0. It s lear that the dstrbuton-funton solutons for a twostage system wth expedted orderng are more omplated than for a four-stage Clark Sarf system see ef. [2]. Ths s manly due to the last term n q. of the omputatonal algorthm, whh further ouples eah stage s regular order and expedted order desons between two onseutve perods. Ths proess of dervng dstrbuton-funton soluton an be ontnued for s and s for all. The expresson naturally beomes more and more omplated as nreases. POPOSITION 2: For = 2,..., N, s f. POOF: eall that s s the soluton of + y y<s s f and only + α [ y D y D s ] = 0. We frst show that f, then s s. Suppose, then for any y s, the frst two terms on the left hand sde of are nonpostve due to the onvexty of y and the optmalty of s ; the thrd term s equal to 0. Thus, beause the left hand sde of s nreasng n y, we onlude that s s. We next prove that, f <, then s < s. Substtutng y by s n, we obtan + y s <s + α [ s D s D s ] = > 0. So agan, beause the left hand sde of s nreasng n y, we must have s <s. The ntuton behnd ths result s as follows. eall that = αk k, whh an be regarded as the ost savng of stage by usng a regular order of one unt from stage n a perod nstead of expedtng one unt next perod; = k k +h s the relatve ost of stage between expedtng one unt from stage + that an be used mmedately for stage and usng regular shppng that arrves next perod. From the perspetve of the whole system, f the former ost savng s hgher than the latter extra ost, then stage should keep some unts avalable to meet regular orderng from stage,.e., s s 0; otherwse, t should not. Ths result further provdes the ondton under whh the optmal expedted base-stok levels are monotone wth the stage ndex: If > for all, then sn s N s. Ths result an smplfy the dstrbuton-funton soluton. For example, f 2, then0 s smplfed to 2 + α P D y s αh + pp D y s, D2 >y = 0. POPOSITION 3: For = 2,..., N, f then s s. POOF: Sne s s s F α, s the soluton of + α [ + y D y D<s y D s ] = 0, to show s s, t s suffent to prove +α[ + s D s D s ] 0. Note that for any possble sample path D = d 0, s d 0. So f + α Ps D s 0, or P D s s α, then the result s vald. The rato 0 /α salways satsfed by the defntons of and. The prevous results not only provde some strutural propertes of the optmal base-stok levels but, more mportantly, wll also be used n the dervaton of bounds for the optmal base-stok levels and omputatonal heursts n the followng setons. Naval esearh Logsts DOI 0.002/nav

6 76 Naval esearh Logsts, Vol We end ths seton wth the followng proposton that presents the omparatve stats results of the optmal basestok levels. POPOSITION 4: s s dereasng n for, ndependent of for >, nreasng n for <, ndependent of for and nreasng n p. s s dereasng n for, ndependent of for >, nreasng n for, ndependent of for > and nreasng n p. Proposton 4 shows that the mpats of system ost parameters on s and s are smlar. Ths proposton s ntutve, as these two base-stok levels are omplementary to eah other. The result an be explaned by the followng lnes of reasonng. For eah stage, f ts relatve expedtng ost gets hgher, then stage s less wllng to use expedton, resultng n a lower expedted base-stok level s. Smlarly, f the relatve expedtng ost of stage, <, beomes hgher, then less expedted shppng would be used at stage, wth the result that the ehelon expedted base-stok level of stage beomes lower. However, f the relatve regular orderng ost of stage rses the atual unt regular shppng ost s lower or the atual expedted shppng ost s hgher, then stage would use more regular shppng, and hene, the regular ehelon base-stok level would beome hgher. Thus, f stage s, <, relatve regular orderng ost beomes hgher, then stage would tend to keep a hgher regular base-stok level, leadng ndretly to an nrease n both the ehelon expedted and regular base-stok levels at stage. That s s dereasng n, <s beause, when the expedtng ost of stage gets hgher, fewer expedted orders would be plaed and, as a result, stage would try not to keep as muh nventory, leadng to a lower ehelon base-stok level for regular shppng. Fnally, that both s and s nrease wth p has ts ntutve appeal: Wth a hgher shortage ost, eah stage should keep a hgher ehelon nventory level to avod shortage. 4. LOW AND UPP BOUNDS In ths seton, we develop several sets of newsvendor-type lower and upper bounds for the optmal ehelon base-stok levels. Before presentng the results, we frst outlne the bas deas used n developng upper and lower bounds. Note that s s determned by y = 0 and y s an nreasng funton of y. If we an fnd a smple upper bound funton ḡy suh that y ḡy, then the soluton of ḡy = 0 s a lower bound for s. Smlarly, f we an fnd another smple lower bound funton gy suh that y gy, then the soluton of gy = 0 s an upper bound for s. The bounds for s an be analogously onstruted. Moreover, the smpler and tghter the upper bound funton ḡy lower bound gy s to y, the smpler and better the resultng lower upper bound. Hene, the hallenge s to fnd smple and tght boundng funtons ḡy and gy. Ths dea was also adopted by Chao and Zhou [2] to derve bounds of the optmal base-stok levels for the Clark Sarf model. However, the addton of expedton opton n eah stage makes the onstruton and dervaton of gy and ḡy more omplex and hallengng here. Sne s s known n a losed form, we shall only develop bounds for s, 2, and for s,. Let 0 = 0. Note that, for =, 2,..., N, 0 and α 0, sne = αk +h > 0 and α = αk + αh 0. Before we present the bounds, we gve the followng result that spefes ondtons under whh stage would never use the expedted shppng mode. POPOSITION 5: For =,..., N, f + α >H + p, then s = ; f α >H +p, then s = and s = for ; Therefore, n the followng dervaton of bounds, we assume + α H + p and α H + p for all. We frst present three sets of newsvendor-type lower bounds. THOM : For =,..., N, the lower bounds for s and s are, respetvely, s and s { = max F, H + p F α }, 2 α H + p { = max F + H + p, F + α + }. 3 α H + p As prevously noted, the rato wthn eah par of large parentheses s learly non-negatve. If the rato s greater than, the resultng lower bound s trval,.e., t s. Naval esearh Logsts DOI 0.002/nav

7 Zhou and Chao: Newsvendor Bounds and Heursts 77 We llustrate the proof of Theorem usng the frst term n the brakets of s and s 2. Applyng the dea explaned at the outset of ths seton, we need to fnd upper bound funtons ḡy for y and 2 y. Note that y = + y y<s +α [ y D s ] + y y<s + y s = + y = + H + ppd > y =ḡy, 4 where the nequalty follows from 0 α < and the onvexty of y. Hene, the soluton of ḡy = 0or F + /H + p s a lower bound for s. Now onsder s2.wehave 2 y = 2 + y s 2 + y s + y s 2 + H + ppd > y 2 = H + ppd > y =ḡy, where the frst nequalty follows from y s 0 and the seond nequalty follows from 4. Thus, the soluton of ḡy = 0or F 2 /H + p s a lower bound for s2. The omplete proof an be done by mathematal nduton, whh we provde n the Appendx. Before we provde some ntuton behnd the dervaton of ths set of lower bounds, we frst defne y s and y s as the margnal ndued penalty ost wthn and between stages respetvely; and [ y D s ] as the expeted margnal ost from the next perod due to the order of the urrent perod. To derve s, we gnore one perod demand to amplfy the margnal ost nrement of the regular order of ths perod to the next, so that stage tends to keep a lower regular base-stok level. For s, we essentally mpose an addtonal margnal ost y s on the expedted order, and stage therefore would set a lower expedted base-stok level. On the bass of Theorem, we an develop another set of lower bounds. We frst defne, for =,..., N, A, = + B,, =,...,, 5 B, = αa,, =,...,, 6 n whh A, = 0 and A, = max{ A,,0}. The omputaton of A, and B, s as follows. Frst, A, = 0 and B, s omputed from 6. Indutvely, suppose A, and B, have been omputed for a gven and all. Then, usng B, and 5 we an ompute A +, for =,...,, and A +,+ = 0. And usng 6, we an ompute B +, for =,..., +. THOM 2: If α α H + p, for = 2,..., N, then, s 2 = max { F k } A, k+ k+ α, k = 2,..., 7 s a lower bound for s, and for =,..., N, s 2 = max { F k+ s a lower bound for s. B, k+ k+ α +, } k =,..., 8 Note that the terms n the brakets of 7 and 8 atually present a sequene of lower bounds for eah stage. To obtan ths sequene of lower bounds for s, we drop y s sequentally from = to = n 6. In other words, we use a smaller ndued penalty ost from downstream stage whh results n a lower base-stok level. For s 2, we use the same dea as that for dervng s. As for the ratos n eah par of parentheses, we an show they are always less than. If the rato s negatve, the resultng lower bound s. For eah stage, by gnorng the ndued penalty ost y s wthn that stage and nreasng the margnal ost nrement from the next perod due to the regular order of the urrent perod, stage, therefore, would set lower basestok levels. Ths dea s appled to derve the followng set of lower bounds. eall that α for all, so the ratos n the next theorem are always between 0 and. THOM 3: For = 2,..., N, f 0, then s 3 = s + F α 9 s a lower bound for s. And for =,..., N, s 3 = s + F α s a lower bound for s. 20 Naval esearh Logsts DOI 0.002/nav

8 78 Naval esearh Logsts, Vol Proposton 3 shows that f, a lower bound for s s s. Theorem 3 provdes a sharper lower bound for s under ths ondton. Although ths bound depends on the optmal s and s for the lower bound of s, we an use the largest avalable lower bounds for s and s to obtan newsvendor-type bounds for s 3 and s 3. Furthermore, note from Proposton 3 that, unless, the lower bounds for s annot be wrtten as the lower bound of the optmal expedted base-stok level of ts downstream plus a non-negatve number. However, from Proposton, the lower bound of s an always be wrtten as the sum of s and a non-negatve number. We have presented three sets of lower bounds for the optmal expedted and regular base-stok levels. These lower bounds do not have a domnatng relatonshp. That s, any lower bound an be a better one, dependng on the problem nstane. We wll provde some dsusson on the performane of dfferent bounds n the numeral studes seton. We next present three sets of newsvendor-type upper bounds by onstrutng dfferent lower bound funtons for y and y. The followng set of upper bounds s developed by usng a smaller margnal ndued penalty ost wthn stage y s. THOM 4: For =,..., N, the upper bounds for s and s are, respetvely: and s s = F = F + + α H + p 2 ; 2 α α H + p α. 22 α Note that our assumpton on ost parameters followng Proposton 5 guarantees that the ratos n the newsvendor bounds above are between 0 and. Agan, we use s and s 2 to demonstrate the dervaton of the upper bounds. To derve s, we frst note the followng nequaltes. y = + y y s [ + α y D ] y D s [ + y D ] y D<s [ + α y D ] y D s = + P y D<s H + pp D2 >y, y D<s + α P y D s αh + pp D2 >y, y D s + α H + ppd2 >y, 23 where the frst nequalty follows from that, by the onvexty of y, y [ y s y D ] y D<s. Clearly, settng qn. 23 to 0 gves an upper bound of s. But we an obtan a better bound by observng that y y s = 0 for y s. Hene, as s s from Proposton, we have + α[ y D y D s ] + α αh + ppd2 >y= gy, and solvng gy = 0 we obtan s. To derve s 2, we apply the nequalty 23, 2 y = 2 + y y<s ] 2 [ + + α H + ppd2 >y y<s 2 H + p α PD2 >y= gy, where the seond nequalty follows from + α 0. Thus, the soluton of gy = 0 s an upper bound of s2, whh s s 2. The seond set of upper bounds s obtaned by replang [ y D y D s ] n 6 for stage by [ y D ],.e., redung the expeted margnal ost from the next perod due to the urrent perod s regular order. THOM 5: For = 2,..., N, s 2 s an upper bound for s, and, let s 0 = s + F s 2 α mn {α, α H + p s an upper bound for s. = s 2, 24 = 0, for =,..., N, α } 25 Naval esearh Logsts DOI 0.002/nav

9 Zhou and Chao: Newsvendor Bounds and Heursts 79 Agan, to ompute 25, the avalable smallest upper bound of s s used nstead of the optmal one. In partular, repettve applatons of Theorem 5 yeld the followng bounds. COOLLAY : One set of upper bounds for s and s s s = s = F F l= l= αl l mn {α l, α H + p l αl l mn {α l, α H + p l α α }, }. We now develop another set of upper bounds for the optmal base-stok levels by replang the margnal ndued penalty ost between stages y s wth a smaller one see detals n the Appendx. Let C = C, =,..., N, wth C 0 = 0. s 3 THOM 6: For =,..., N, { = mn F C H + p F 2, s an upper bound for s, and + mn{αc }, C } s 3 = F 2 s an upper bound for s. H + p + αc αh + p As n the ase of lower bounds, none of the upper bounds developed above domnates the others. That s, any one of these upper bounds an be sharper, dependng on the problem nstane. 5. NAL LADTIMS In the prevous setons, the leadtmes for regular and expedted orderng are assumed to be and 0, respetvely. In ths seton, we extend the results to the ase where leadtmes for regular and expedted orderng are l r = l + and l e = l, respetvely, wth l beng an arbtrary non-negatve nteger. Clearly, ths represents an extenson of Fukuda s model [0] to seral mult-ehelon systems. Ths extenson an be obtaned from the model n Seton 3 by nsertng stages to represent unts of leadtme. Spefally, we an represent eah of the l unts of leadtme as an auxlary stage wth only regular shppng mode, zero orderng ost, and zero ehelon holdng ost. Under suh a ost struture, one an expedted or a regular shppng s ntated at a nonauxlary stage, there s no nentve to keep t n the auxlary stages. Mathematally, ths s equvalent to settng the optmal ehelon base-stok levels for these auxlary stages at nfnty. The bottom up reursve algorthm for omputng the optmal ehelon base-stok levels for the more general leadtme ase s as follows. edefne, for =,..., N, Let = k k + α l h, = αk k. y = y + αl H + p[y Dl + ], and let s be the mnmzer of y. For =,..., N, ompute y = y + [ ] y s + α y D s, 28 s = arg mn y y, 29 and for =,..., N, + y = + y + αl + [ y Dl+ s ], 30 s+ = arg mn y + y. 3 The algorthm above an be derved from the followng argument. Frst, note that the results of the prevous setons an be easly extended to the ase where some stages only have one transportaton mode avalable. eall that for the Clark Sarf model wth only one transportaton mode suppose the unt orderng ost s k, =,..., N and leadtmes between stages beng, the optmal ehelon base-stok levels are omputed as follows for a detaled dervaton see ef. [2]. y = αk + αh y + αh + p[y D2 ], Naval esearh Logsts DOI 0.002/nav

10 80 Naval esearh Logsts, Vol and s = arg mn y; for >, y = y s y = αk + αh y + α [ y D], s = arg mn y, n whh s s the optmal base-stok level. Therefore, f, for example, stage only has one transportaton mode wth leadtme, whereas stages and + have two transportaton modes wth leadtmes 0 and, then the optmal ehelon basestok level s for stage s obtaned by optmzng, where s alulated as above by takng as of the prevous setons and s takng value s ; + y s alulated from q. 3, takng as and s as s. We now llustrate how to apply ths dea to derve qs for a seral system wth l e = l and l r = l +, =,..., N. Insert l stages before the orgnal stage, eah of whh only has the regular transportaton mode wth leadtme. We all these l stages added stage, added stage 2,..., added stage l from downstream to upstream. The added stage faes ustomer demand and s alloated a shortage ost p and ehelon holdng ost 0. We denote the added stage by a subsrpt a,. Therefore, a, y = αh + p[y D2 ] wth a mnmzer beng nfnty, and as a result we have 2 a, y = a,y and a,2 y = α[ 2 a, y D] = α2 H + p[y D3 ]. Ths proess ontnues untl the orgnal stage s reahed and we obtan y = y + a,l y = y + αl H + p [ y Dl + ], where the frst equalty s due to zero leadtme for the expedted shppng at stage and the mnmzer of the added stage l s nfnty. It should be noted that we have α l h n sne the holdng ost at orgnal stage wll nur l perods later when the order arrves at the added auxlary stage. Applyng algorthm 3 for the orgnal stage, we obtan y = y + y s + α [ y D s ]. Ths proess ontnues and we obtan that, for stage +, + y = + y + αl + [ y Dl + s ], n whh y s omputed from the added l stage before stage, and + remans to be gven by q. 3. On the bass of the above algorthm, t s lear that Proposton and Propostons 2 4 n Seton 3 ontnue to hold under ths more general leadtme settng. Proposton does not hold n general beause when applyng the analogous argument as the proof of Proposton, s may be greater than s sne the soluton of + [ y Dl s ]=0may be greater than s. In the prevous analyss, however, Dl = 0. We now demonstrate the hanges for eah set of lower and upper bounds under ths more general settng of leadtmes. We omt the detaled dervaton beause t s analogous to the analyss of the prevous setons wth the algorthm Let L, = k= l k. The frst set of lower bounds wth general leadtmes s, for =, 2,..., N s and s = max = max F L, + F L, + F L, + F L, + + αl +, α L, H + p + αl +,+ α L,+ H + p αl +, α L, H + p, αl +,+ + α L,+ H + p,,, where t s lear that the rato wthn eah par of parentheses s non-negatve from the defnton of and. Agan, f the rato s larger than, from our prevous defnton, the lower bound s. To generalze the seond set of lower bounds presented n Theorems 2, we frst need to redefne for =,..., N, A, = + α l B,, =,...,, B, = αa,, =,...,, n whh A, = 0 and A, = max{ A,,0}. If αl +,+ αl,+ H + p, for = 2,..., N, then, s 2 = max { F L, +k A, k+ k+ α L +,+, } k = 2,..., Naval esearh Logsts DOI 0.002/nav

11 Zhou and Chao: Newsvendor Bounds and Heursts 8 s a lower bound for s, and for =,..., N, { s 2 = max F L, +k+ B, k+ k+ α L +,+ + k =,..., The last set of lower bounds here s, for =,..., N, α s 3 = s + F l l + α l +. The expresson of s 3 s the same as 20 beause the leadtme dfferene between the expedted and regular shppng s stll. The upper bounds wth general leadtmes need to be modfed more arefully beause we an no longer apply part of Proposton as we dd n the prevous dervaton. Consder the frst set of upper bounds. As s may not be less than s, the upper bound for s beomes, for =, 2,..., N s = F L, + α L, H + p., αl α For the upper bound s,ass s s stll true, we have s = F L, ++, α α α L, H + p., αl α For the seond set of the upper bounds, s 2 s no longer vald. But we stll have s 2, whh s modfed as follows. s 2 = s + F l + α α α L, H + p, αl α }. Fnally, for the last set of upper bounds, redefne C = α l C, =,..., N, wth C 0 = 0 and C = max{ C,0}.For =,..., N, and s 3 s 3 = F L, + C, α L, H + p = F + αc L, +2. α L,+ H + p Note that we no longer have the seond term n the brakets of 26 for the last set of upper bounds. 6. HUISTICS AND NUMICAL STUDIS In ths seton, we develop a smple heurst for the optmal ehelon base-stok levels based on the lower and upper bounds obtaned n the preedng setons. We also ondut a numeral study to demonstrate the effetveness of the heurst. We frst fous on the ase wth l e = 0. For omputatonal purposes, the demand dstrbuton s assumed to be dsrete n ths seton. For =, 2,..., N, let s s { = max } { } s, =, 2, 3, = max s s, =, 2, 3 ; { } { } = mn s, =, 2, 3, s = mn s, =, 2, 3. It s lear that s s s and s s s, for =,..., N. Furthermore, t follows from Proposton 2 that f, then s s. Hene, n the followng, f s > s, then we set s = s ; and f, we set s >s s =. s For =, 2..., N and 0 β, set s h = s h = [ ] + β s βs, [ ] + β s βs, n whh []s the round off operator. We hoose β = 0.5 as the heurst poly. The heurst poly works n exatly the same manner as the orgnal top down ehelon base-stok poly, but s h and s h are used as the ehelon base-stok levels for stage. In the followng, we ondut a numeral study to test the effetveness of the heurst. We onsder a three-stage N = 3 system. The system parameters for the examples are p {30, 60}, h {0., }, k {4, 0}, k {2, 6}, for =, 2, 3, and α = We present two groups of numeral examples lassfed by the demand dstrbutons. We use the relatve error on the optmal system ost as the measure for the effetveness of the heurst. Let x = x,..., x N wth x beng the ntal ehelon nventory level at stage. Denote vx and ˆvx as the optmal ost and the ost of the heurst poly wth a gven x, respetvely. To alulate these osts, we use suessve approxmaton wth a planng horzon T = 00, and we observe that the total dsounted ost onverges n our numeral results. The relatve error of the heurst s defned as rror% = max 200 x 200,=,...,N { ˆvx vx vx } 00%, n whh, to avod the nfluene of the ntal state on the performane of the heurst, we onsder a reasonable large Naval esearh Logsts DOI 0.002/nav

12 82 Naval esearh Logsts, Vol Fgure. Performane summary of the heurst: l e = le 2 = le 3 = 0. [Color fgure an be vewed n the onlne ssue, whh s avalable at number of possble ombnatons of ntal ehelon nventory levels,.e., x [ 200, 200] for all. Fgure reports the dstrbuton of errors wth respet to the number of nstanes for these two groups of examples. In roup, demand follows Posson dstrbuton wth parameter λ {5, 0, 50}. By restrtng k >k, we generate 432 nstanes by dfferent ombnatons of the system parameters for eah demand rate. The average relatve error among 432 nstanes for λ = 5 s 0.57% wth the maxmum 3.06%, for λ = 0 s 0.52% wth the maxmum 4.28%, and for λ = 50 s 0.33% wth the maxmum.70%. The average relatve error for all 296 nstanes s 0.47%. From these results, t an be seen that the average performane of the heurst gets better when λ nreases. The demand dstrbuton for the seond group of numeral examples s Negatve Bnomal wth four sets of dfferent mean and varane 30, 40, 30, 20, 6, 8, and 6, 24. Ths allows us to observe the mpat of demand varane on the performane of the heurst. The oeffent of varaton for eah set s 0.2, 0.37, 0.47, and 0.82 orrespondngly. ah par of demand mean and varane wth dfferent system ost ombnatons also generates four sets of numeral examples and eah set nludes 432 nstanes. The average relatve error among 432 nstanes for the frst set s 0.37% wth the maxmum 2.65%, for the seond s 0.42% wth the maxmum 3.62%, for the thrd s 0.48% wth the maxmum 2.88%, and for the fourth s 0.49% wth the maxmum 2.64%. The average relatve error for all 728 nstanes s 0.44%. We observe that the average performane of the heurst s better wth a smaller oeffent of varaton. Ths result s ntutve, as a devaton from the optmal soluton would ause a larger ost f the demand s more varable. For the system n whh l e = l and l r = l + wth l > 0, a heurst an be smlarly derved one we have the bounds presented n Seton 5. To see how the resultng heurst β = 0.5 performs, we test the preedng two groups of examples exept that the leadtmes are now l e = 3, le 2 =, and l e 3 = 3 for the expedted orders and lr = l e +. When demand s Posson, for eah of the three sets of nstanes dfferentated by the demand rate, the average and maxmum errors are 0.56%, 2.%, 0.42%,.57%, and 0.20 %,.0%. When demand s Negatve Bnomal, the average and maxmum errors for eah of the four sets of problems are summarzed as: 0.38%,.27%; 0.28%,.24%; 0.43%, Fgure 2. Performane summary of the heurst: l e = le 3 = 3, le 2 =. [Color fgure an be vewed n the onlne ssue, whh s avalable at Naval esearh Logsts DOI 0.002/nav

13 Zhou and Chao: Newsvendor Bounds and Heursts 83.44%; and 0.58%, 2.67%. These numeral results show that the heurst also works well under ths more general leadtme settng. We report the dstrbuton of errors wth respet to the number of nstanes n Fg. 2. We onlude ths seton wth some observatons from our numeral results on the relatve performane of dfferent lower and upper bounds reported n Seton 4. We fnd that, for lower bounds, n most ases, the frst set of lower bound s for expedted order and the thrd set of lower bound s 3 for regular order serve as the best lower bounds n our numeral examples. For nstane, among 432 nstanes of the thrd set of roup 2 examples, 432 of are from s s 2 and 44 of are 2 s 3 from s 3 ; 324 of are from s s 3, 339 of are from s s 3 2, and of are from s s 3 3. Wth respet to upper bounds, the frst 3 set of upper bound s and the thrd set of upper bound s 3 for expedted base-stok levels show a smlar performane, and are better than the seond set of upper bound s 2 ; for the regular base-stok level, the frst set of upper bound s s the most effetve one. But n general, no lear pattern seems to emerge on the mpat of system parameters on bounds. We also test the effetveness of bounds wth dfferent value of α. We fnd that when α gets smaller, s 3 and s 3 get better loser to the optmal ones, s 3 beomes worse. 7. CONCLUSION In ths artle, we study an nfnte-horzon, perodrevew, seral produton/nventory system wth expedted and regular shppng modes avalable between stages. We derve strutural propertes of the optmal poles and develop newsvendor-type lower and upper bounds for the optmal ehelon base-stok levels. Among the dfferent sets of lower and upper bounds, some perform better than others under dfferent system parameters. These bounds lead to a smple and effetve heurst for the optmal nventory ontrol poly. Numeral studes show that the heurst performs well. We generalze the omputatonal algorthm and other results to the ase where expedted and regular orders n eah stage have leadtmes l and l +, respetvely, for an arbtrary non-negatve nteger l. The bounds and heurst poles are gven n losed forms n terms of umulatve demand dstrbuton funtons and prmtve system parameters, hene, they mmedately reveal the mpat of the system parameters on the ontrol poles. We beleve that these results shed lghts on the struture of the optmal poles for mult-ehelon seral nventory systems wth dual shppng modes, and on ther mplementablty. APPNDIX In ths appendx, we gve the proofs for Propostons 4, 5, and Theorems 6. In these proofs, the exhange of expetaton and dervatve s ustfed by Lebnz s rule. POOF OF POPOSITION 4: In ths proof, we denote the densty funton of a gener one-perod demand by f. It follows from the defnton and the onvexty of y and y that s s dereasng n and s s nreasng n. To show that s s dereasng n, t suffes to prove that y s nreasng n. To that end, let y = 0 be wrtten as gy, = 0, and let s be ts soluton. eall that gy, s nreasng n y. Suppose.Ifgy, s nreasng n, then g s, g s, = 0. Hene, t follows from gy, s nreasng n y that, s, determned by gy, = 0, satsfes s s. Ths shows that s s dereasng n. Note that g y, = + y y s y s + α y t df t. 0 Notng y, y =, where y, y represents the ross dervatve wth respetve to y and, we obtan g y, = y<s + y y<s y<s + αf y s α s f y s s = y<s + y y<s y<s + αf y s 0, where the seond equalty follows from s beng the mnmzer of, and the nequalty follows from y 0 when y<s, and y<s 0 beause s s dereasng n. We next show that both s and s are dereasng n for <. Suppose y y, 0 for, then for +, frst take dervatve of wth respet to y, y y s = + α y ξ df ξ and then take dervatve wth respet to, 0 y y s y, = α y ξ y, df ξ 0, 0 whh mples that s s dereasng n for <.Fors +, whh s the soluton of + y = + + y = 0 and from the prevous analyss, t s lear that + y y, = y y, 0, whh mples that s+ s dereasng n for <. Naval esearh Logsts DOI 0.002/nav

14 84 Naval esearh Logsts, Vol As s s ndependent of, we frst prove s 2 s nreasng n. From q. 0, we have 2 y y, = < 0 whh mples that s2 s nreasng n. Now suppose y y, for <, then for s, and y y, y s = α y ξ y, dfξ < y y, = y y, < 0. < 0 Therefore, both s and s are nreasng n for <. The proof of the result that both s and s are nreasng n p follows smlar steps and we leave t for nterested reader. POOF OF POPOSITION 5: Frst note that, f α > H + p, then k + k α >H + p for k. To show part, we need to frst show that for all, y H + p α PD>y. 32 If q. 32 s vald and f + α >H + p for some, then y > 0, and so, s =. Moreover, from q. 32 y = + y y s + α [ y D ] y D s + [ y D ] y D<s + α [ y D ] y D s + P y D<s + α P y D s H + p α P D + >y, y D<s α H + p α P D + >y, y D s + α H + p PD + >y α where the frst nequalty follows from the fat that y [ y s y D ] y D<s. Therefore, f α >H + p, y > 0 and s =. Moreover, as + α = α α >H +p, sk = for k. And from the pror argument, s k = for k. Now, we prove q. 32 by nduton. Frst, for =, q. 32 s equalty. Suppose t s true for. For +, note that + y = + + y y<s + [ + + α H + p + H + p α ] PD + >y y<s α PD + >y 33 where the frst nequalty follows from the nduton assumpton and the seond nequalty follows from that + α 0. So the proof s ompleted. POOF OF THOM : Sne s and s are the solutons of y = 0 and y = 0, respetvely, to prove the results, t s suffent to show that for all, and y y + H + ppd > y, 34 H + ppd > y, 35 y α α H + ppd > y, 36 y + α + α H + ppd > y. 37 We prove these nequaltes by nduton. Consder qs. 34 and 35 frst. q. 34 s learly true for s. We have shown q. 35 wth = n Seton 3. Assumng qs. 34 and 35 hold for, we next prove that they hold for +. Frst, from qs. 4, an analogous dea as n the proof of 2 and the ndutve assumpton, + y = + + y s + + y + + H + ppd > y + = H + ppd > y. We then prove q. 35 for +. Note that + y = + + y + y<s + + α [ y ] + D y D s [ y ] + y<s + + y + y s H + ppd > y, Naval esearh Logsts DOI 0.002/nav

15 Zhou and Chao: Newsvendor Bounds and Heursts 85 where the frst nequalty follows from the onvexty of + y and 0 α<and the seond nequalty s from the ndutve assumpton. We proeed to show qs. 36 and 37. The nequalty q. 36 s learly true for =. For q. 37 and =, y = + y [ y<s + α ] y D s + α y y<s + α y s = + α αh + ppd > y, where the nequalty follows from the onvexty of. Assumng qs. 36 and 37 hold for, we prove that they hold for +. Frst, for q. 36, + y + + y + + α + α H + ppd > y + = α + α H + ppd>y, where the seond nequalty follows from the ndutve assumpton of q. 37. We fnally prove q. 37 for +. Note that For =,..., N, from Theorem, y = + y y<s + α [ y ] D y D s + α [ y ] D y D s [ + α α P y D s α H + pp D2 >y, y D s ] [ + α α P y D s α ] P D2 >y, y D s [ = + α α P y D s, D2 y ] + α + PD2 y = B, + α + PD2 y, + y = + + y + y<s + + α [ y ] + D y D s α y + y<s + + α [ y ] + D y D s α y + y<s + + α y + y s α +2 α + H + ppd > y, where the nequaltes agan follow from the onvexty of + 0 α<. Hene, the proof s ompleted. and where the thrd nequalty follows from the assumpton that α α H + p. Ths valdates the ase k = for q. 39. The dervaton above mples that + y = + + y y<s + + y + + A +, + α + PD2 y α + PD2 y. POOF OF THOM 2: For qs. 7 and 8, we need to show for = 2,..., N, y k+ A, k+ + α PDk y, and =,..., N, y B, k+ k = 2,...,, 38 k+ + α + PDk+ y, k =,...,. 39 Ths proves the ase for k = 2 of q. 38. On the bass of these, suppose q. 38 holds for some k =, then for q. 39, y + α [ y ] D y D s +α A, + P y D s + + l= α l l l P D + y, y D s + B, + + α l+ l l PD + y, l= Naval esearh Logsts DOI 0.002/nav

16 86 Naval esearh Logsts, Vol thus q. 39 holds for k = +. Meanwhle, + y = + + y y<s + + y + + αa, + + α l+ l l PD + y l= + = A +, + + α l+ l l PD + y, l= whh verfes that q. 38 holds for k = +. So, we omplete the nduton proof. and POOF OF THOM 3: It suffes to show that for =, 2,..., N, y + α P D y s, 40 y + α P D y s. 4 For q. 40, note that y = + y y<s + y [ + α y D ] y D s + α P y D s where the last nequalty follows from that y. The nequalty q. 4 follows from y [ + α y D ] y D s Therefore, the theorem s proved. + α P y D s. POOF OF THOM 4: eall the nequalty q. 32. As s s, the soluton of + + α 2 H + p α PD>y = 0 must be an upper bound of s,.e., s = F + α H + p 2 α s. To see ths, note that f s <s then by q. 33, we have on y s, y + + α 2 H + p α PD>y, hene, s 0, and t mples s s ; On the other hand, f s s, then the result holds automatally beause s s. We next show q. 22. Note that y y s = 0ony s. Hene, as s s and for y s, from q. 32, y [ = + α y D ] y D s + α α H + p α PD + >y, 42 whh mples q. 22. To verfy q. 22 s ndeed an upper bound for s, we stll need to show t s greater than or equal to s. Ths s true, sne pluggng y = s n q. 42 shows that the rght hand sde s negatve, mplyng that q. 22 s at least as large as s. POOF OF THOM 5: The valdty of q. 24 follows from Proposton. For q. 25, note that [ y D ] = + [ y D ] y D<s + + α 2 H + p α D > y y D<s + H + p α D > y y D<s H + p α P D>y s, where the frst nequalty follows from q. 42, the seond one follows from + α 0 and the last one from H + p α 0. Moreover, as s s and for y s, y = + α [ y D ] y D s + α [ y D ] + α α H + p α P D>y s, 43 whh mples the seond term n the brakets of q. 25. It an be shown that the soluton of q. 43 s greater than or equal to sne pluggng y = s n the equatons above shows that the rght hand sde of q. 43 s negatve. For the frst term n q. 25, note that for y>s, y = + α [ y D ] y D s + α [ y D ] y D s + α P y D s = + α α P y D<s, where the frst nequalty follows from that y 0 for y s and s s and the seond nequalty follows from the defnton of y for y>s. Therefore, q. 25 s vald. Naval esearh Logsts DOI 0.002/nav